Regulation of renewable resource exploitation

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Regulation of renewable resource exploitation

Idris Kharroubi, Thomas Lim, Thibaut Mastrolia

To cite this version:

Idris Kharroubi, Thomas Lim, Thibaut Mastrolia. Regulation of renewable resource exploitation.

SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019,

58 (1), pp.551-579. �10.1137/19M1265740�. �hal-03013860�

Regulation of renewable resource exploitation

Idris Kharroubi

Thomas Lim

Thibaut Mastrolia

November 25, 2019

Abstract

We investigate the impact of a regulation policy imposed on an agent exploiting a pos- sibly renewable natural resource. We adopt a principal-agent model in which the Principal looks for a contract, i.e.taxes/compensations, leading the Agent to a certain level of ex- ploitation. For a given contract, we first describe the Agent’s optimal harvest using the BSDE theory. Under regularity and boundedness assumptions on the coefficients, we ex- press almost optimal contracts as solutions to HJB equations. We then extend the result to coefficients with less regularity and logistic dynamics for the natural resource. We end by numerical examples to illustrate the impact of the regulation in our model.

Key words: Contract Theory, BSDEs, HJB PDE, Logistic SDE.

1 Introduction

The exploitation of natural resources is fundamental for the survival and development of the growing human population. However, natural resources are limited since they are either non renewable (e.g. minerals, oil, gas and coal) so that the available quantity is limited, or renewable (e.g. food, water and forests) and in this case the natural resource is limited by its ability to renew itself. In particular, an excessive exploitation of such resources might lead to their extinc- tions and therefore affect the depending economies with, for instance, high increases of prices and higher uncertainty on the future. Thus, the natural resource manager faces a dilemma:

either harvesting intensively the resource to increase her incomes, or taking into account the potential externalities induced by an overexploitation of the resource and impacting her future ability to harvest the resource. It has been nevertheless emphasized in [7] that in some cases it is optimal for natural resource manager to harvest until the extinction of the resource. This optimal harvesting strategy thus leads to costs for the global welfare related to the environment degradation.

Therefore, the management and the monitoring of the exploitation of natural resources are a balance between optimal harvest for the natural resource manager and ecological implications for public organizations. This second issue has attracted a lot of interest, especially from governance institutions. For example in its last annual report on sustainable development, the statistical office of the European Union Eurostat dedicates a full section to the question of sustainable consumption and production (see [11], Section 12).

The management of natural resources have also attracted a lot of interest from the academic community. Many studies on natural resources exploitation tried to describe the possible effect of economic incentives on the exploitation (see e.g. [5, 14, 27, 15]). These references stress the need of an incentive policy to ensure the sustainability of the resource. However, even if

∗Sorbonne Universit´e, LPSM,idris.kharroubi @upmc.fr.

†Laboratoire d’Analyse et Probabilit´es, Universit´e d’Evry-Val d’Essonne and ENSIIE,lim@ensiie.fr.

‡CMAP, Ecole Polytechnique, IP Paris,thibaut.mastrolia@polytechnique.edu. This author acknowledges the financial supports of the Chaire Financial Risks hosted by the Louis Bachelier Institute and the ANR project PACMAN ANR-16-CE05-0027.

the regulator have access to the abundance level of natural resource, the unobservability of the natural resource manager behavior induces moral hazards. Thus, the regulator’s issue is to incentivize the resource manager to optimally reduce the cost of the resource degradation, together with ensuring a minimal incomes for the manager, under moral hazard. To the best of our knowledge, this question has been addressed only in the discrete-time framework (see for instance [13]) without considering any randomness in the dynamics of the resource. The aim of this work is to investigate this problem in continuous time with randomness in the system.

To deal with this issue, we consider a principal-agent model under moral hazard. The first elements of contract theory with moral hazard appeared in the 60’s with the articles [3, 30] in which the mechanisms of controlled management were investigated. Then, it has been extended and named asagency problem (see among others [29, 24]) by considering discrete-time models.

Concerning the continuous-time framework, the agency problem with moral hazard has been first studied in [17] by modelling the uncertainty of risky incomes with a Brownian motion.

The agency problem can be roughly described as follows. We associate a moral hazard problem with a Stackelberg game in which the leader (named the Principal) proposes at time 0 a compensation to the follower (named the Agent) given at a maturity T > 0 fixed by the contract, to manage the wealth of the leader. Moreover, the Principal has to propose a compensation high enough (called the reservation utility) to ensure a certain level of utility for the Agent. Although the Principal cannot directly observe the action of the Agent, the former can anticipate the best reaction effort of the latter with respect to a fixed compensation. Hence the agency problem remains to design an optimal compensation proposed by the Principal to the Agent given all the constraints mentioned above under moral hazard.

The common approach to solve this problem consists in proceeding in two steps. The first step is to compute the optimal reaction of the Agent given a fixed compensation proposed by the Principal, i.e. solving the utility maximization problem of the Agent. In all the papers mentioned above, the shape of considered contracts is fundamental to solve the Agent problem by assuming that the compensation is composed by

– a constant part depending on the reservation utility of the Agent, – a part indexed by the (risky) incomes of the Principal,

– the certain equivalent gain of utility appearing in the Agent maximization.

Using the theory of Backward Stochastic Differential Equations (BSDE for short), [9] proved that this class ofsmooth contracts, having a relevant economic interpretation, is not restrictive to solve the agency problem. The second step consists in solving the Principal problem. Taking into account this optimal reaction of the Agent, the goal is to compute the optimal compensa- tion. As emphasized in [26] and then in [8, 9], this problem remains to a (classical) stochastic control problem with the wealth of the Principal and the continuation utility value of the Agent as state variables.

In this paper, we identify the natural resource manager as the Agent. The Principal refers to a regulator, which can be a public institution that monitors the resource manager’s activities.

The resource manager can either harvest or renew the natural resource. In the first case the production is sold at a given price on the market and in the second case the resource manager pays for each unit of renewed natural resource. To regulate the natural resource exploitation, the Principal imposes a tax/compensation to the Agent depending on the remaining level of resource at the terminal time horizon. We suppose here that the Agent is risk-averse and we model his preference with an exponential utility function¹. For a given harvesting strategy, the Agent total gain is composed by the cumulated amounts paid/earned by renewing/harvesting the natural resource and the regulation compensation/tax. The Agent’s aim is then to maximize the expected utility of his total gain over possible harvesting strategies.

On the other side, given the previous optimal harvest of the Agent, the regulator aims at fixing a tax/compensation policy that incentives the Agent to let a reasonable remaining level

1See for instance [4] for more details on this kind of utility function and the economical interpretations of it.

of natural resource. As a public institution, we assume that the regulator is risk-neutral.

The main features to model the dynamic of a renewable natural resource are its birth and death rates and the inter-species competition. Besides, due to random evolution of the popu- lation, we consider uncertainty in the available abundance. Following [12, 2, 22] we choose to model the evolution of the natural resource by a stochastic logistic diffusion.

We then focus on the Principal-agent problem. We first characterize the Agent behavior for a fixed regulation policy represented by a random variableξ. Following the BSDEs approaches to deal with exponential utility maximization, we get a unique optimal harvesting strategy as a function of the Z component of the solution to a quadratic BSDE with terminal condition ξ (see [25, 18]).

We next turn to the regulator problem which consists of maximizing an expected terminal reward depending on the regulation taxξand the level of remaining natural resource according to the Agent’s optimal response. By writing the explicit form of the resource manager’s optimal strategy, we turn the regulator problem into a Markov stochastic control problem of a diffusion with controlled drift. We then look for a regular solution to the related PDE to proceed by verification. However, in our case we face the following three issues.

– By considering the logistic dynamics for the resource abundance population, the HJB PDE related to the Principal problem involves a term of the form x²∂_xv wherexstands for the resource population abundance andv is the Principals value function. This term, induced by the inter-species competition in the classical logistic case, prevents us from using existence results of regular solutions to PDEs.

– The shape of the optimal harvest of the manager leads to irregular coefficients for the related PDE, which also prevents from getting regular solutions.

– Due to the exponential preferences of the Agent, the Principal’s admissible strategies need to satisfy an exponential integrability condition. However, the linear preferences of the Principal leads to an optimal contract that is not necessarily exponential integrable.

Therefore, the regulator problem might not have an optimal regulation policy.

To deal with these issues, we first study a model for which the inter-species competition coeffi- cientµof the population is bounded. Hence, the termx²∂_xvis replaced byxµ(x)∂_xv. We then construct a regular approximation of the Hamiltonian. By considering the related PDE, we de- rive a regular solution (see Proposition 4.1) together with an almost optimal control satisfying the admissibility condition (see Theorem 4.2). We notice that our approach can be related to that of Fleming and Soner [28], which consists of an approximation of the value function by a sequence of smooth value functions to derive a dynamic programming principle. We next turn to the logistic casei.e. µ(x) =xfor which we show that the almost optimal strategy obtained for a truncation of µ remains an almost optimal strategy for a large value of the truncation parameter (see Theorem 4.3).

We finally illustrate our results by numerical experiments. We compute the almost optimal strategies using approximations of solutions to HJB PDEs and show that the regulation has a significant effect on the level of remaining natural resource.

The remainder of the paper is the following. In Section 2 we describe the considered mathe- matical problem. We then solve in Section 3 the manager’s problem for a given regulation policy.

In Section 4, we first provide almost optimal strategies in the case where the coefficient µ is bounded and we extend our result to the logisitic dynamics. We end Section 4 by economical insights and numerical experiments.

Notations and spaces

We give in this part all the notations used in this paper. Let (Ω,F,P) be a complete probability space. We assume that this space is equipped with a standard Brownian motion W and we denote byF:= (Ft)_t≥0its right-continuous and complete natural filtration.

Letp≥1 and a time horizonT >0, we introduce the following spaces

– P(R) (resp. Pr(R)) will denote the σ-algebra of R-progressively mesurable, F-adapted (resp. F-predictable) integrable processes.

– S_T^p is the set of processesX,P(R)-mesurable and continuous satisfying E[ sup

0≤t≤T

|Xt|^p]<+∞. – H^pT is the set of processesX,Pr(R)-mesurable satisfying

E h Z T

0

|Xt|²dt^p₂i

<+∞.

– For an integerq≥0, a subsetD ofR^q and for anyν ∈(0,1), we denote byC^1+ν(D) the set of continuously differentiable functions f : D→Rsuch that

|f|1= sup

x∈D

|f(x)|+ X

1≤i≤q

|∂x_if(x)|+ sup

x,y∈D

X

1≤i≤q

|∂x_if(x)−∂x_if(y)|

|x−y|^ν

<∞,

and by C^2+ν(D) the set of twice continuously differentiable functions f : D →R such that

|f|2+ν = sup

x∈D

|f(x)|+ X

1≤i≤q

|∂xif(x)|+ X

1≤i,j≤q

|∂xi,xjf(x)|

+ sup

x,y∈D

X

1≤i,j≤q

|∂x_i,x_jf(x)−∂x_i,x_jf(y)|

|x−y|^ν < ∞.

2 The model

2.1 The natural resource

We fix a deterministic time horizonT >0 and we suppose that the natural resource abundance X_t^µ at timet≥0 is given by

X_t^µ=X₀+ Z t

0

X_s^µ(λ−µ(X_s^µ))ds+ Z t

0

σX_s^µdW_s, t∈[0, T], (2.1) where X0, λ andσ are positive constants. The quantities X0 and λcorrespond to the initial natural resource abundance and the growth rate respectively. The map µrepresents the com- petition inside the species considered or more generally an auto-degradation parameter for a natural resource. We assume that the mapµsatisfies the following assumption

(H0)µis a map fromR+ toR+ such that (2.1) admits a unique strong solution inS_T². Note that Assumption(H0)holds for instance if the mapx7−→xµ(x) is Lipschitz continuous.

Another important example is the so-called logistic equation where µ(x) = xon R+, see for example in [12]. In this last case, SDE (2.1) admits an explicit unique solution that will be denoted in the sequel byX and given by

X_t= X₀e^{(λ−σ22)t+σWt} 1 +X0Rt

0e^{(λ−σ22)s+σWs}ds

, t∈[0, T].

The ecological interpretation of this model is the following. At timet, if the coefficientµ(X_t^µ) is larger thanλthen the drift of the diffusion is negative. Therefore the abundance of the natural resource X_t^µ decreases in mean. Conversely, if µ(X_t^µ) is smaller thanλ then the drift of the diffusion is positive. Hence, the abundance X_t^µ increases in mean. For more details see for instance [22, Proposition 3.4].

More general models can be used in practice and one of the main challenges, see [22], is to rely branching processes with birth and death intensities to the solutions of continuous SDEs.

2.2 The Agent’s problem

We consider an agent who tries to make profit from the natural resource. We suppose that this agent owns facilities to either harvest or renew this resource. We assume that his action happends continuously in time and we denote byα_this intervention rate at timet,i.e. the abun- danceX_t^µwill decrease of an amountα_tX_t^µper unit of time. This means that if the intervention rateα_tis positive (resp. negative), the Agent harvests (resp. renews) the natural resource. We denote byA the set ofF-adapted processes defined on [0, T] and valued in [−M , M] whereM and M are two nonnegative constants. If the Agent is prohibited to renew the resource then M = 0. This setAis called the set of admissible actions.

To take into account the control α of the Agent on the natural resource abundance, we introduce the probability measureP^α defined by its densityH^αw.r.t. Pgiven by

dP^α dP

_F

T

:=H_T^α, where the processH^αis defined by

H_t^α:= exp

− Z t

0

αs

σ dWs−1 2

Z s

0

αs

σ

2

ds

, t∈[0, T].

In the sequel, we denote byE^α andE^αt the expectation and conditional expectation given F_t respectively, for anyt∈[0, T], under the probability measureP^α.

Forα∈ A, we get from Girsanov Theorem (see e.g. Theorem 5.1 in [19]) that the process W^α defined by

W_t^α:=Wt+ Z t

0

α_s

σds , t∈[0, T],

is a Brownian motion under the probabilityP^α. Thus, for a given admissible effortα∈ A, the dynamics ofX can be rewritten under the probabilityP^αas

X_t^µ=x+ Z t

0

X_s^µ(λ−µ(X_s^µ))−αsX_s^µ ds+

Z t

0

σX_s^µdW_s^α, t∈[0, T].

This new dynamics reflects the evolution of the population with a rate α_t per unit of time.

Hence,αtX_t^µ has to be seen as the speed of the exploitation of the natural resource at timet.

Remark 2.1. We choose a harvesting/renewing component of the form αX^µ. This allows to take into account not only the impact of the effort of the agent on the system but also the abundance of the resource itself. We note that this form of strategy is a particular case of the form presented in [16, (2.4)]. From a technical point of view, this particular form is due to the presence of a quadratic cost (see (2.2)) and allows to derive optimal efforts under a closed form. Possible extensions which are let for further research are to consider more general forms including finite variations and jumps as in the dynamic proposed in [16].

We then are given a price function p: R⁺ →R⁺ and we suppose that the price per unit of the natural resource on the market is given byp(X_t^µ) at timet≥0. We make the following assumption on the price functionp.

(Hp)There exists a constantP such thatp(x)x≤P for allx∈[0,+∞).

This price functionpallows to take into account the dependence w.r.t. the abundance (the more abundant the resource is, the cheaper it will be and conversely). Such a price dependence has already been used to model liquidity effects on financial market, where empirical studies showed that the impact is of the formp(x) =P e^−β1xβ2,x∈R+, for some positive constantsP, β₁andβ₂(see e.g. [1, 21]). In particular,(H_p)is satisfied for this type of dependence. Another basic example for which(H_p)holds is the casep(x) =P x⁻¹,x >0. This last example reflects the inability to buy the natural resource once it is extinct.

We assume that the manager sells the harvested resource on the market at pricep(X_t^µ) per unit at timet ifαtis positive, and pays the pricep(X_t^µ) per unit of natural resource at timet ifαt is negative to renew this one. This provides the global amountRT

0 p(X_t^µ)X_t^µαtdtover the time horizon [0, T].

We also suppose that giving an effort is costly for the manager and we consider the classical quadratic cost functionk:R→R+ given byk(α) = ^|α|₂², α∈R. Thus, the Agent is penalized by the instantaneous amountk(α_t) per unit of time for a given effort α∈ A. This leads to the global paymentRT

0 k(α_t)dtover the considered time horizon [0, T].

In our investigation, we recall that the activity of the natural resource manager is regulated by an institution (usually an environment administration) who is taking care about the size of the remaining natural resource. To avoid an over-exploitation, the regulator imposes a tax on the Agent depending on the remaining resource. This tax amount is represented by an FT-measurable random variable ξand is paid at timeT. Note thatξ can be either positive or negative. In this last case, it means that the regulator gives a compensation to the manager.

Throughout the paper we assume that the Agent’s preferences are given by the exponential utility functionu_A defined by

uA(x) :=−exp −γx

, x∈R,

where γ is a positive constant corresponding to the risk aversion of the Agent. We define the value functionVA(ξ) of the Agent associated to the taxation policyξby

V_A(ξ) := sup

α∈AE^α

h−exp

−γ Z T

0

p(X_s^µ)X_s^µα_sds− Z T

0

|αs|²

2 ds−ξi

. (2.2)

For a fixed taxξ, we denote byA^∗(ξ) the set of effortsα^∗∈ Asatisfying the following equality E^α

∗h

−exp

−γ Z T

0

p(X_s^µ)X_s^µα^∗_sds− Z T

0

|α^∗_s|²

2 ds−ξi

=VA(ξ). An effortα^∗∈ A^∗(ξ) is said to be optimal for the fixed taxξ.

2.3 The Principal’s problem

The aim of the regulator is to stabilize the resource population at a fixed target size at the maturityT. For that, a taxξis chosen to incentivize the Agent to manage the natural resource so that the remaining population is close to the targeted size. Hence, the regulator benefits from the tax paid by the Agent and is penalized through a cost functionf depending on the size of the resource at maturityT. The expected reward under the action α∈ A of the Agent is then given by

E^α

ξ−f(X_T^µ) .

Typically, we have in mind f(x) =c(β−x)⁺ meaning that the regulator targets a population size β > 0 at timeT for the sustainability of the resource and pays the costc per unit if the

natural resource is over-consumed. This functionf can be seen as the amount that the regulator must pay to reintroduce the missing resource.

We suppose that the resource manager is rational. Therefore, the Principal anticipates that for a taxξ, the Agent will choose an effortαin the setA^∗(ξ). Note that this set is not necessarily reduced to a singleton², hence, as usual in moral hazard problems (see for instance [17] for the formulation of the moral hazard problem), the regulator solves

sup

ξ

V^P(ξ), withV^P(ξ) = sup

α∈A^∗(ξ)

E^α h

ξ−f(X_T^µ)i

, (2.3)

whereξ lives in a set of suitable contracts defined in the following section.

2.4 Class of contracts and utility reservation

We now introduce a reserve utility R which is a negative constant. This reserve means that the regulator cannot penalize too strongly the Agent for economical reasons so that the utility VA(ξ) expected by the Agent has to be greater thanR. For instance, we can chooseRsuch that the regulator monitors the Agent by promising the same expected utility as the case without regulation (see Section 4.3.1 for more details). This example reflects a non-punitive taxation policy in which the regulator purely monitors the activities of the Agent. In our model, the sign of the tax ξis on purpose. This means that the natural resource manager pays the fee to the regulator whenξis positive and conversely, the regulator compensates the Agent’s activity when ξ is negative. Moreover, we need to impose an exponential integrability on the tax ξ to ensure the well-posdness of VA(ξ). We therefore introduce the class C_R^µ of admissible taxes defined as the set ofFT-measurable random variablesξsuch that

VA(ξ)≥R , (2.4)

and there exists a constantγ⁰>2γsuch that E

exp(γ⁰|ξ|)

<+∞. (2.5)

This last condition is very convenient since it allows to deal with the problem by using the theory of BSDEs. Moreover, a straightforward application of Cauchy-Schwarz inequality ensures that the optimization problemsV_A(ξ) andV^P(ξ) take finite values.

3 Optimal effort of the natural resource’s manager

We first solve the optimal problem of the Agent (2.2) under taxation policyξ∈ C^µ_R. As in [9], the following result shows that solving the Agent problem gives both an optimal effortα^∗ and a particular representation of the taxξwith respect to the solution of a BSDE.

Theorem 3.1. Let ξ ∈ C^µ_R and Assumption (Hp) be satisfied. There exists a unique pair (Y0, Z)∈(−∞,R]˜ ×H²T withR˜ :=^log(−R)_γ such that

(i) the tax has the following decomposition ξ=Y0−

Z T

0

g(X_t^µ, Zt) +σ² 2 γ|Zt|²

dt+ Z T

0

σZtdWt, (3.6) whereg is defined for any(x, z)∈R+×Rby

g(x, z) = |a^∗(x, z)|²

2 −p(x)xa^∗(x, z)−a^∗(x, z)z , and

a^∗(x, z) = (p(x)x+z)∨(−M)

∧M , (3.7)

2In our investigation, we will show that the setA^∗(ξ) is reduced to a single element.

(ii) the value of the Agent is given by

V_A(ξ) =−exp(γY₀),

(iii) the process α^∗(ξ) defined by α^∗_t(ξ) = a^∗(X_t^µ, Z_t) is the unique optimal effort associated with the taxξ given by (3.6).

Proof. The proof is divided in three steps and is related to the BSDE associated with the value function of the Agent. We first introduce a dynamic extension of the optimization problem (2.2). We denote byJ(t, ξ) the dynamic value function of the Agent at timetfor a taxξwhich is defined by

J(t, ξ) := ess inf

α∈A E^αt

h exp

−γ Z T

t

p(X_s^µ)X_s^µαsds− Z T

t

k(αs)ds−ξi . Note thatV_A(ξ) =−J(0, ξ).

Step 1. Dynamic utility of the Agent and BSDE.We characterizeJ(·, ξ) as the unique solution of a BSDE and we derive the optimal control by using comparison results.

Letα∈ A, we introduce the processJ^α(ξ) defined by J_t^α(ξ) :=E^P

α

t

h exp

−γ Z T

t

p(X_s^µ)X_s^µαsds− Z T

t

k(αs)ds−ξi , so that

J(t, ξ) := ess inf

α∈At

J_t^α(ξ). (3.8)

Step 1a. Martingale representation and integrability.

We know that the processH_t^α(exp(γRt

0 k(αs)−p(X_s^µ)X_s^µαs

ds)J_t^α(ξ))_0≤t≤T is a (P,F)-martingale.

In view of the condition (2.5), there exists ε >0 and q >1 such that (2 +ε)qγ ≤γ⁰. Hence, forp >1 such that ¹_p+¹_q = 1, sinceαis bounded and Condition (2.5) is satisfied, we get from H¨older’s inequality

E[|H^α_TJ_T^α(ξ)|^2+ε] ≤ E[|H^α_T|^(2+ε)p]^1pE[|e^(2+ε)qγξ|]^1q < +∞, whereH^α_t :=H_t^αexp

γRt

0 k(αs)−p(X_s^µ)X_s^µαs

ds

. Hence, by using Doob’s maximal inequal- ity,H^αJ^α(ξ)∈ S^2+ε. So by using the martingale representation theorem, we know there exists a processZ^α∈H^2+εT such that

H^α_tJ_t^α(ξ) = J₀^α+ Z t

0

σZ^α_sdWs, t∈[0, T]. Therefore,J^αsatisfies

dJ_t^α(ξ) = (αtZ˜_t^α−γ(k(αt)−p(X_t^µ)X_t^µαt)J_t^α(ξ))dt+σZ˜_t^αdWt, where ˜Z_t^α= ^Z

α t

H^α_t +J_t^αα_σ2^t for anyt∈[0, T], andJ_T^α(ξ) = exp(γξ).

We now prove that ˜Z^α ∈ H²T. From (2.5), the boundedness of α and Assumption (H_p), there exists a positive constantC >0 such that

E hZ T

0

|Z^α_t

H^α_t +J_t^ααt

σ²|²dti

≤ 2 E

hZ T

0

|Z^α_t H^α_t|²dti

+E hZ T

0

|J_t^ααt

σ²|²dti

≤ C 1 +E

hZ T

0

|Z^α_t H^α_t|²dti

.

We set ˜q:= 1 +^ε₂ and ˜p >1 such that ¹_p˜+¹_q˜= 1. Using H¨older and BDG Inequalities and since Z^α∈H^2+εT , we get

E[ Z T

0

|Z^α_t

H^α_t |²dt] ≤ E[ sup

t∈[0,T]

|(H^α_t)⁻¹|² Z T

0

|Z^α_t|²dt]

≤ E[ sup

t∈[0,T]

|(H^α_t)⁻¹|^{2 ˜p}]^p1˜E[ Z T

0

|Z^α_t|²dt^q˜ ]^1q˜

< +∞. Consequently, we get ˜Z^α∈H²T.

Step 1b. Comparison of BSDEs and optimal effort. We now turn to the characterization of the solution to (3.8) by a BSDE. We introduce the following BSDE

dJ_t(ξ) = − inf

a∈[−M ,M]

G(X_t^µ, J_t(ξ),Z˜_t, a)dt+σZ˜_tdW_t, J_T(ξ) = exp(γξ), (3.9) where

G(x, j,z, a)˜ := γ(k(a)−p(x)xa)j−a˜z .

This BSDE has a Lipschitz generator and square integrable terminal condition from (2.5).

Therefore it admits a unique solution in S²×H²T. Moreover, for any α∈ A, we notice that (J^α(ξ),Z˜^α) satisfies the following BSDE

dJ_t^α(ξ) = −G(X_t^µ, J_t^α(ξ),Z˜_t^α, αt)dt+σZ˜_t^αdWt, J_T^α(ξ) = exp(γξ). By classical comparison Theorem (see for instance [10, Theorem 2.2]), we have

J_t(ξ) ≤ J(t, ξ), ∀t∈[0, T]. Then, we notice that BSDE (3.9) can be rewritten

dJ_t(ξ) = −G

X_t^µ, J_t(ξ),Z˜_t, a^∗ X_t^µ, Z˜_t γJ_t(ξ)

dt+σZ˜_tdWt, J_T(ξ) = exp(γξ).

In particular, we have J(ξ) = J^{a∗ Xµ,}

Z˜ γJ(ξ)

(ξ) by uniqueness of the solution to BSDE (3.9).

Therefore, we get

J_t(ξ) = J(t, ξ) and a^∗ X^µ, Z˜ γJ(ξ)

∈ A^∗(ξ). (3.10)

We now prove that this optimal effort is unique. Let ˜α∈ Abe another optimal effort, then we have

J₀^α˜ = J^a

∗ X^µ,_γJ(ξ)^Z˜

0 .

From strict comparison Theorem (see again [10, Theorem 2.2]) we getJ^α˜=J^{a∗ Xµ,}

Z˜ γJ(ξ)

and G

X^µ, J(ξ),Z, a˜ ^∗ X^µ, Z˜ γJ(ξ)

= G

X^µ, J(ξ),Z,˜ α˜

, dt⊗dP−a.e.

By the uniqueness of the minimizer ofG(X_t^µ, J_t(ξ),Z˜_t,·),we deduce that ˜α=a^∗ X^µ,_γJ(ξ)^Z˜ . Step 2. Representation ofξ.

Since, by definition, the processJ^{a∗ Xµ,}

Z˜ γJ(ξ)

is positive, we can define the processesY andZ by

Y :=

log

J^{a∗ Xµ,}

Z˜ γJ(ξ)

γ and Z :=

Z˜ γJ^{a∗ Xµ,}

Z˜ γJ(ξ)

. (3.11)

We obtain

dYt=− k(a^∗(X^µ, Zt))−p(X_t^µ)X_t^µa^∗(X^µ, Zt)−a^∗(X^µ, Zt)Zt+σ² 2 γ|Zt|²

dt+σZtdWt, YT =ξ .

We first proveY ∈ S². Note for anyt∈[0, T], by using Jensen inequality, we have 1

γlog(J_t(ξ)) ≥ E^α

∗

t

hZ T

t

k(α^∗_s)−p(X_s^µ)X_s^µα^∗_s ds+ξi

≥ E^α

∗

t [ξ]−T P M , whereα^∗ stands fora^∗ X^µ,_γJ(ξ)^Z˜

andM =M ∨M. We then notice – ifJ_t(ξ)≥1 we have

0 ≤ log(J_t(ξ)) ≤ J_t(ξ), – if 0≤J_t(ξ)<1 we have

1

γlog(J_t(ξ))

≤ T P M+E^α

∗

t [|ξ|]. Hence, there exists a constantC >0 such that

1

γlog(J_t(ξ))

2

≤ C(1 +E^α

∗

t [|ξ|]²) + 1

γ²|J_t(ξ)|², t∈[0, T]. From Young inequality, we get

E h

sup

t∈[0,T]

1

γlog(J_t(ξ))

2i

≤ 2C 1 +E

h sup

t∈[0,T]

H_T^α∗ H_t^α∗

4i

+E[|ξ|⁴] + 1

γ²E h

sup

t∈[0,T]

|J_t(ξ)|²i .

Sinceα^∗ is bounded, we haveE

hsup_t∈[0,T]Hα∗ T

H_t^α∗

⁴i

<+∞. UsingJ(ξ)∈ S², we obtain E[ sup

t∈[0,T]

1

γlog(J_t(ξ))

2

]<+∞. Which impliesY ∈ S².

We now checkZ ∈ H²T. To this end, we use a localization procedure by introducing the sequence of stopping times (τ_n)_n≥1defined by

τ_n := infn

t∈[0, T], Z t

0

|Z_s|²ds≥no

∧T ,

for anyn≥1. Similarly to the proof of [6, Theorem 2], we apply Itˆo’s Formula toι(|Y|) where ι(x) =_γ¹2(e^γx−γx−1) forx∈R. We obtain

ι(|Y0|) = ι(|Yτ_n|) + Z τn

0

ι⁰(|Ys|)sgn(Ys) g(X_s^µ, Zs) +σ²γ 2 |Zs|²

−1

2ι⁰⁰(|Ys|)σ²|Zs|² ds

− Z τ_n

0

σι⁰(|Ys|)sgn(Ys)ZsdWs.

Sinceι⁰⁰−γι⁰ = 1 andι⁰(x)≥0 for x≥0, we get from BDG and Young inequalities E

hZ τ_n

0

|Zs|²dsi

≤ C 1 +E

h sup

t∈[0,T]

e^γ|Yt|+ Z T

0

e^γ|Yt| 1 +|Ys|i

. (3.12)

From the definition of Y and sinceα^∗ is bounded, there exists a constantC such that 2γ|Y_t| ≤ C+ 2γE^α

∗

t [|ξ|].

Using Jensen and H¨older inequalities we get another constantC⁰ such that E

h sup

t∈[0,T]

e^2γ|Yt|i

≤C⁰E h

sup

t∈[0,T]

H_T^α∗ H_t^α∗

^γ

0 γ0 −γi^{γ0 −γ}_γ0

E h

e^2γ0|ξ|i_γ^γ0

.

Sinceα^∗ is bounded, we haveE h

sup_t∈[0,T]Hα∗ T

H_t^α∗

^γ

0 γ0 −γi

<+∞and we get from (2.5) E

h sup

t∈[0,T]

e^2γ|Yt|i

< +∞. Sending nto ∞in (3.12), we get from Fatou’s LemmaZ∈H².

Step 3. Conclusion. We directly deduce (ii) and (iii) from (3.11) together with (3.10) given

that (i) has been proved in Step 2.

4 The problem of the regulator

In this section, we focus on the regulation policy. In view of (2.3) and Theorem 3.1 the regula- tor’s problem turns to be

V_R^P = sup

ξ∈C^µ_R

E^α

∗(ξ)[ξ−f(X_T^µ)]. (4.13)

We first provide almost optimal contracts for a bounded parameterµby a PDE approach. We then extend the study to the logistic case withµ(x) =x.

4.1 Almost optimal strategies for bounded auto-degradation and cost parameters

We introduce the following class of contracts Ξ^µ := n

Y_T^Y0,Z,µ=Y0− Z T

0

h(X_t^µ, Zt) +σ² 2 γ|Zt|²

dt+ Z T

0

σZtdWt, Y0≤R , Z˜ ∈ Zo

, (4.14)

whereZ denotes the subset of predictable processes ofH²T such that

E[exp(γ⁰|Y_T^Y0,Z,µ|)] < +∞, (4.15) for someγ⁰ >2γand we recall that ˜R= ^log(−R)_γ . Whenµis the identity, we omit the exponent µin the previous definitions.

From Theorem 3.1, constraint (2.4) and integrability conditions (2.5) and (4.15), the set C_R^µ coincides with Ξ^µ so that the regulator’s problem (4.13) becomes

V_R^P = sup

Y₀≤R, Z∈Z˜

E^a

∗(X^µ,Z)[Y_T^Y0,Z,µ−f(X_T^µ)], (4.16)

with

Y_t^Y0,Z,µ = Y₀− Z t

0

k(α^∗_s)−p(X_s^µ)X_s^µa^∗(X_s^µ, Z_s) +σ² 2 γ|Z_s|²

dt+ Z t

0

σZ_sdW_s^∗ , t∈[0, T], where W^∗ stands for W^a∗(Xµ,Z). We notice that the function to maximize in V_R^P is non- deacreasing w.r.t. the variableY0. Therefore the constraintY0≤R˜ is saturated and (4.16) can be rewritten under the following form

V_R^P = sup

Z∈ZE^a

∗(X^µ,Z)[Y_T^R,Z,µ˜ −f(X_T^µ)]. (4.17) To construct a solution to the problem (4.17), we introduce the related HJB PDE given by

( −∂tv−H

x, ∂_xv(t, x), ∂_xxv(t, x)

= 0, (t, x)∈[0, T)×R^∗+, v(T, x) =−f(x), x∈R^∗+,

(4.18) where the HamiltonianH is given by

H(x, δ₁, δ₂) = sup

z∈R

xp(x)a^∗(x, z)−k(a^∗(x, z))−σ²

2 γz²+x(λ−µ(x)−a^∗(x, z))δ₁

+σ²

2 x²δ2, (x, δ1, δ2)∈R^∗+×R×R,

anda^∗is given by (3.7). We first extend PDE (4.18) to the whole domain [0, T]×Rby considering the change of variablew(t, y) :=v(t, e^y) for any (t, y)∈[0, T]×R. We get the following PDE

( −∂_tw− H

y, ∂_yw(t, y), ∂_yyw(t, y)

= 0, (t, y)∈[0, T)×R, w(T, y) =−f(e^y), y∈R,

(4.19) where

H(y, δ₁, δ₂) := sup

z∈R

e^yp(e^y)a^∗(e^y, z)−a^∗(e^y, z)² 2 −σ²

2 γz²+ (λ−σ²

2 −µ(e^y)−a^∗(e^y, z))δ₁

+σ²

2 δ₂, (y, δ₁, δ₂)∈R×R×R.

Our aim is to construct a regular solution to this PDE to proceed by verification. Unfortunately, the coefficients of PDE (4.19) are not smooth enough to do so. To overcome this issue, we provide a smooth approximationHεofHfor which we get regular solutions.

Moreover, we introduce the following assumption, which ensure that the optimal control derived from the PDE satisfies the admissibility condition,i.e. belongs toZ.

(H’)There existsν ∈(0,1) such that

(i) the mapy7→µ(e^y) belongs toC^1+ν(R), (ii) the mapy7→f(e^y) belongs toC^2+ν(R), (iii) the mapy7→p(e^y)e^y belongs toC^1+ν(R).

Proposition 4.1. Under (H’), there exists a family{Hε, ε >0} of functions from R³ to R such that the PDE

( −∂twε− Hε

y, ∂ywε(t, y), ∂yywε(t, y)

= 0, (t, y)∈[0, T)×R, wε(T, y) =−f(e^y), y∈R,

(4.20) admits a unique solutionw_ε inC^2+ν([0, T]×R) and

sup

R³

H − H_ε

≤ ε (4.21)

for any ε >0.

The proof of Proposition 4.1 consists in an approximation by regularization of the original HamiltonianH. As it is quite technical we postpone this proof to the appendix.

We are now able to describe almost optimal contracts and related almost optimal efforts using the functionsw_εgiven by Proposition 4.1.

Theorem 4.2. Suppose that (H’)holds. For any ε >0, the tax policyξ_εgiven by ξ_ε= ˜R−

Z T

0

g(X_t^µ, Z_t^ε) +1

2σ²γ|Z_t^ε|²+Z_t^ε(λ−µ(X_t^µ)) dt+

Z T

0

Z_t^ε X_t^µdX_t^µ , where

Z_t^ε = −∂xwε(t,log(X_t^µ))

1 +γσ² , t∈[0, T], (4.22)

is2T ε-optimal for the regulator problem:

V_R^P ≤ E^a

∗(X^µ,Z^ε)

ξε−f(X_T^µ)

+ 2T ε .

Proof. We fix some control Z ∈ Z and we apply Itˆo’s formula to the process Y_t^R,Z,µ˜ + wε(t,log(X_t^µ))

t∈[0,T]

Y_T^R,Z,µ˜ +wε(T,log(X_T^µ)) = R˜+wε(0,log(X0)) +

Z T

0

∂twε(s,log(X_s^µ)) +(λ−σ²

2 −µ(X_s^µ)−a^∗(X_s^µ, Zs))∂xwε(s,log(X_s^µ)) +p(X_s^µ)a^∗(X_s^µ, Z_s)X_s^µ−k(a^∗(X_s^µ, Z_s))−σ²

2 γ|Zs|² +σ²

2 ∂xxwε(s,log(X_s^µ)) ds +σ

Z T

0

(∂xwε(s,log(X_s^µ)) +Zs)dW_s^∗, whereW^∗ stands forW^a∗(Xµ,Z). Sincew_ε∈C^2+ν([0, T]×R) andZ ∈ Z we get

E^a

∗(X^µ,Z)h

Y_T^R,Z,µ˜ +w_ε(T,log(X_T^µ))i

≤ R˜+w_ε(0,log(X₀)) +

Z T

0

E^a

∗(X^µ,Z)h

∂twε+H

., ∂ywε, ∂yywε

(s,log(X_s^µ))i ds . From (4.21) we get

E^a

∗(X^µ,Z)h

Y_T^R,Z,µ˜ +wε(T,log(X_T^µ))i

≤ R˜+wε(0,log(X0)) +T ε +

Z T

0

E^a

∗(X^µ,Z)h

∂_tw_ε+Hε

., ∂_yw_ε, ∂_yyw_ε

(s,log(X_s^µ))i ds ,

and sincewε is solution to (4.20), we get E^a

∗(X^µ,Z)h

Y_T^R,Z,µ˜ −f(X_T^µ)i

≤ R˜+wε(0,log(X0)) +T ε . SinceZ is arbitrarily chosen in Z we get

V_R^P ≤ R˜+w_ε(0,log(X₀)) +T ε . (4.23)

We now take Z = Z^ε where Z^ε is given by (4.22). We now notice that Z^ε ∈ Z since Z^ε is bounded, and by definition of Z^εwe have

H

log(X^µ), ∂ywε(.,log(X^µ)), ∂yywε(.,log(X^µ))

= X^µp(X^µ)a^∗(X^µ, Z^ε)−a^∗(X^µ, Z^ε)²

2 −σ²

2 γ Z^ε

2+σ²

2 ∂yywε(.,log(X^µ)) + λ−σ²

2 −µ(X^µ)−a^∗(X^µ, Z^ε)

∂_yw_ε(.,log(X^µ))

for any [0, T]. A straightforward application of Itˆo’s formula and Girsanov Theorem give E^a

∗(X^µ,Z^ε)[Y_T^{R,Z˜ ε,µ}−f(X_T^µ)] = R˜+w_ε(0,log(X₀)) +

Z T

0

E^a

∗(X^µ,Z^ε)h

∂_tw_ε+H

., ∂_yw_ε, ∂_yyw_ε

(s,log(X_s^µ))i ds . From Propositions (4.20) and (4.21) we get

E^a

∗(X^µ,Z^ε)[Y_T^{R,Z˜ ε,µ}−f(X_T^µ)] ≥ R˜+wε(0,log(X0))−T ε . Hence, we get from (4.23)

V_R^P ≤ E^a

∗(X^µ,Z^ε)

[Y_T^{R,Z˜ ε,µ}−f(X_T^µ)] + 2T ε . Therefore, we getξ_ε=Y_T^{R,Z˜ ε,µ} is a 2T ε-optimal policy for the regulator.

4.2 Extension to the logistic equation and continuous cost

We consider in this section an approximation method to build a sequence of almost optimal taxes in the case the classical logistic dynamic for SDE (2.1), i.e. µ(x) = x. More precisely, we introduce a sequence of approximated models from which we derive almost optimal strategy from the previous section. We show that this sequence remains almost optimal for the logistic model. We also weaken the assumption(H’)(ii) as follows.

(Hf’)The function f is bounded and continuous onR.

We introduce the sequence of mollifiersρn: R→R,n≥1, defined by ρ_n(x) := nρ(nx)

R

Rρ(u)du , x∈R, where the functionρ: R→Ris defined by

ρ(x) := exp −1 1− |x|²

1|x]<1. We then define the functionsfn,n≥1, by

f_n(x) :=

Z

R

f(y)ρ_n(x−y)dy , x∈R.

From classical results, we know thatfn satisfies(H’)(ii) for alln≥1 andfn converges tof as ngoes to infinity uniformly on every compact subset ofR.

We also define the functionsµ_n: R→R,n≥1, by

µ_n(x) := x Θ(x+eⁿ+ 1)−Θ(x−(eⁿ+ 1))

, x∈R, (4.24)

where the function Θ :R→Ris given by Θ(u) :=

Ru

−∞ρ(r)dr R

Rρ(r)dr , u∈R. (4.25)